Editing Mode (statistics) (section)

{{short description|Value that appears most often in a set of data}}
{{For|the music theory concept of "modes"|Mode (music)|}}
{{pp-pc}}
In [[statistics]], the '''mode''' is the value that appears most often in a set of data values.<ref>[[Damodar N. Gujarati]]. ''Essentials of Econometrics''. McGraw-Hill Irwin. 3rd edition, 2006: p. 110.</ref> If {{mvar|'''X'''}} is a discrete random variable, the mode is the value {{mvar|x}} at which the [[probability mass function]] takes its maximum value (i.e., {{math|1=''x'' = argmax<sub>''x''<sub>''i''</sub></sub> P('''''X''''' = ''x''<sub>''i''</sub>)}}). In other words, it is the value that is most likely to be sampled.

Like the statistical [[mean]] and [[median]], the mode is a way of expressing, in a (usually) single number, important information about a [[random variable]] or a [[population (statistics)|population]]. The numerical value of the mode is the same as that of the mean and median in a [[normal distribution]], and it may be very different in highly [[skewed distribution]]s.

The mode is not necessarily unique in a given [[discrete distribution]] since the probability mass function may take the same maximum value at several points {{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, etc. The most extreme case occurs in [[Uniform distribution (discrete)|uniform distributions]], where all values occur equally frequently.

A mode of a [[continuous probability distribution]] is often considered to be any value {{mvar|x}} at which its [[probability density function]] has a locally maximum value.<ref name=Zhang2003>{{cite journal | last1 = Zhang | first1 = C | last2 = Mapes | first2 = BE | last3 = Soden | first3 = BJ | year = 2003 | title = Bimodality in tropical water vapour | journal = Q. J. R. Meteorol. Soc. | volume = 129 | issue = 594 | pages = 2847–2866 | doi = 10.1256/qj.02.166 | bibcode = 2003QJRMS.129.2847Z | s2cid = 17153773 }}</ref> When the probability density function of a [[continuous distribution]] has multiple [[local maximum|local maxima]] it is common to refer to all of the local maxima as modes of the distribution, so any peak is a mode. Such a continuous distribution is called [[multimodal distribution|multimodal]] (as opposed to [[unimodal distribution|unimodal]]).

In [[Symmetric distribution|symmetric]] [[unimodality|unimodal]] distributions, such as the [[normal distribution]], the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric unimodal distribution, the sample mean can be used as an estimate of the population mode.